Uniform continuity over locally compact quantum groups

We define, for a locally compact quantum group $G$ in the sense of Kustermans--Vaes, the space of $LUC(G)$ of left uniformly continuous elements in $L^\infty(G)$. This definition covers both the usual left uniformly continuous functions on a locally compact group and Granirer's uniformly continuous functionals on the Fourier algebra. We show that $LUC(G)$ is an operator system containing the $C^*$-algebra $C_0(G)$ and contained in its multiplier algebra $M(C_0(G))$. We use this to partially answer an open problem by Bedos--Tuset: if $G$ is co-amenable, then the existence of a left invariant mean on $M(C_0(G))$ is sufficient for $G$ to be amenable. Furthermore, we study the space $WAP(G)$ of weakly almost periodic elements of $L^\infty(G)$: it is a closed operator system in $L^\infty(G)$ containing $C_0(G)$ and--for co-amenable $G$--contained in $LUC(G)$. Finally, we show that--under certain conditions, which are always satisfied if $G$ is a group--the operator system $LUC(G)$ is a $C^*$-algebra.

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